Well, I would say that the only kind of rights capable of creating moral obligations on others are moral rights, and moral rights are the flip side of moral responsibilities. In general I have a moral right to do whatever I have a moral responsibility to do. Moral responsibilities come ultimately from God, either immediately through divine law or mediately through nature.
For example, I have a moral responsibility, which arises from nature, to protect and take care of my own life. Therefore I have a moral right to live and obtain the things necessary to continue my life. Hence I have a right not to be killed by you, and as such, it is immoral for you to kill me.
It seems to me that the difference between us is that you would locate the ultimate source of these rights in natural, universal human desire, whereas I locate the ultimate source of these rights in God.
The problem with your position is that overwhelming and/or collective desire by itself cannot establish a moral norm. I can show you this if you just agree, and I hope you will agree, with this statement: "Moral rules are rules about what choices should be made by humans".
For if moral rules concern what choices should be made by humans, well, we make choices with our wills (whether free or not). Examining the nature of the will should therefore tell us about the rules for making choices.
If it's true that all humans universally desire X, then it would seem to be a moral norm that X is morally good. The problem comes in when humans universally desire not only X but also Y, and that X and Y turn out to be contradictory goals in a non-obvious way.
In such a case, we would want to say that X and Y are both moral imperatives, but it would turn out that pursuing X would lead away from Y. In fact, every individual goal of any desire must be in at least a bit of conflict with every other goal, if only for the reason that our time on earth is finite, and hence, giving time toward the attainment of any one goal must necessarily imply losing that time for the attainment of any other goal.
So in practice, there must be a hierarchy of goals which the will can choose from, with one and only one thing at the apex: the very highest and greatest possible desire, for the sake of which it would be worth while to spend all of one's time pursuing and neglect every other goal.
Or, at least, it has to be that way IF you want to have a moral system that is absolute.
If you also want your moral system to be universal, it needs to be the case that every human being desires the same thing as their highest goal. And if you ALSO want your moral system to be objective, there needs to be something objective about human nature that makes it the case that humans are just the kind of things that desire that particular goal by nature.
So, you can build a moral system on human desire, but unless you construct a hierarchy of values, show that exactly one value is at the very top, and that this follows from something objective about human nature, you can't ultimately escape subjectivism.